Graph Search Chris Amato Northeastern University Some images and slides are used from: Rob Platt, CS188 UC Berkeley, AIMA
What is graph search?
Graph search: find a path from start to goal
– what are the states?
– what are the actions (transitions)?
– how is this a graph?
Start state Goal state
What is graph search?
Graph search: find a path from start to goal
– what are the states?
– what are the actions (transitions)?
– how is this a graph?
Start state Goal state
What is graph search?
Graph search: find a path from start to goal
– what are the states?
– what are the actions (transitions)?
– how is this a graph?
Start state
Goal state
What is graph search?
Graph search: find a path from start to goal
– what are the states?
– what are the actions (transitions)?
– how is this a graph?
Start state
Goal state
What is a graph?
Graph:
Edges:
Vertices:
Directed graph
What is a graph?
Graph:
Edges:
Vertices:
Undirected graph
What is a graph?
Graph:
Edges:
Vertices: Also called states
Also called transitions
Defining a graph: example
Defining a graph: example
How many states?
Defining a graph: example
Defining a graph: example
Pairs of states that are “connected” by one turn of the cube.
Example: Romania
l On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest
l Formulate goal: Be in Bucharest
l Formulate problem: l states: various cities l actions: drive between cities
l Find solution: l sequence of cities, e.g., Arad,
Sibiu, Fagaras, Bucharest
Graph search
Given: a graph, G
Problem: find a path from A to B
– A: start state
– B: goal state
Graph search
Given: a graph, G
Problem: find a path from A to B
– A: start state
– B: goal state
How?
Problem formulation
A problem is defined by four items:
l initial state e.g., “at Arad”
l successor function S(x) = set of action–state pairs
e.g., S(Arad) = {⟨Arad → Zerind, Zerind⟩, . . .}
l goal test, can be explicit, e.g., x = “at Bucharest” implicit, e.g., NoDirt(x)
l path cost (additive)
e.g., sum of distances, number of actions executed, etc. c(x, a, y) is the step cost, assumed to be ≥ 0
l A solution is a sequence of actions leading from the initial state to a goal state
A search tree
Start at A
A search tree
Successors of A
A search tree
Successors of A
parent children
A search tree
Let's expand S next
A search tree
Successors of S
A search tree
A was already visited!
A search tree
A was already visited! So, prune it!
A search tree
In what order should we expand states?
– here, we expanded S, but we could also have expanded Z or T
– different search algorithms expand in different orders
Breadth first search (BFS)
Breadth first search (BFS)
Breadth first search (BFS)
Start node
Breadth first search (BFS)
Breadth first search (BFS)
Breadth first search (BFS)
Breadth first search (BFS)
We're going to maintain a queue called the fringe – initialize the fringe as an empty queue
Fringe
Breadth first search (BFS)
– add A to the fringe
fringe Fringe A
Breadth first search (BFS)
-- remove A from the fringe -- add successors of A to the fringe
fringe
Fringe B C
Breadth first search (BFS)
-- remove B from the fringe -- add successors of B to the fringe
fringe
Fringe C D E
Breadth first search (BFS)
fringe
Fringe D E F G
-- remove C from the fringe -- add successors of C to the fringe
Breadth first search (BFS)
fringe
Fringe D E F G
Which state gets removed next from the fringe?
Breadth first search (BFS)
fringe
Fringe D E F G
Which state gets removed next from the fringe? What kind of a queue is this?
Breadth first search (BFS)
fringe
Fringe D E F G
Which state gets removed next from the fringe? What kind of a queue is this?
FIFO Queue! (first in first out)
Breadth first search (BFS)
Breadth first search (BFS)
What is the purpose of the explored set?
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists?
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n?
– b: branching factor – d: depth of shallowest solution – complexity = ???
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a solution?
– b: branching factor – d: depth of shallowest solution – complexity =
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a solution?
– b: branching factor – d: depth of shallowest solution – complexity =
What is the space complexity of BFS? – how much memory is required?
– complexity = ???
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a solution?
– b: branching factor – d: depth of shallowest solution – complexity =
What is the space complexity of BFS? – how much memory is required?
– complexity =
BFS Properties
Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a solution?
– b: branching factor – d: depth of shallowest solution – complexity =
What is the space complexity of BFS? – how much memory is required?
– complexity = Is BFS optimal? – is it guaranteed to find the best solution (shortest path)?
Uniform Cost Search (UCS)
Uniform Cost Search (UCS)
Notice the distances between cities
Uniform Cost Search (UCS)
Notice the distances between cities – does BFS take these distances into account?
Uniform Cost Search (UCS)
Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage?
Uniform Cost Search (UCS)
Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?
Uniform Cost Search (UCS)
Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?
How do we fix this?
Uniform Cost Search (UCS)
Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?
How do we fix this? UCS!
Uniform Cost Search (UCS)
Same as BFS except: expand node w/ smallest path cost
Length of path
Uniform Cost Search (UCS)
Same as BFS except: expand node w/ smallest path cost
Length of path
Cost of going from state A to B:
Minimum cost of path going from start state to B:
Uniform Cost Search (UCS)
Same as BFS except: expand node w/ smallest path cost
Length of path
Cost of going from state A to B:
Minimum cost of path going from start state to B:
BFS: expands states in order of hops from start UCS: expands states in order of
Uniform Cost Search (UCS)
Same as BFS except: expand node w/ smallest path cost
Length of path
Cost of going from state A to B:
Minimum cost of path going from start state to B:
BFS: expands states in order of hops from start UCS: expands states in order of How?
Uniform Cost Search (UCS)
Simple answer: change the FIFO to a priority queue – the priority of each element in the queue is its path cost.
Uniform Cost Search (UCS)
UCS
Fringe A
Path Cost 0
Explored set:
UCS
140 118 75
Explored set: A
Fringe A S T Z
Path Cost 0 140 118 75
UCS
140 118 75
146
Explored set: A, Z
Fringe A S T Z T
Path Cost 0 140 118 75 146
UCS
140 118 75
146 229
Explored set: A, Z, T
Fringe A S T Z T L
Path Cost 0 140 118 75 146 229
UCS
140 118 75
239 220 146 229
Explored set: A, Z, T, S
Fringe A S T Z T L F R
Path Cost 0 140 118 75 146 229 239 220
UCS
140 118 75
239 220 146 229
Explored set: A, Z, T, S
Fringe A S T Z T L F R
Path Cost 0 140 118 75 146 229 239 220
UCS
140 118 75
239 220
336 317
146 229
Explored set: A, Z, T, S, R
Fringe A S T Z T L F R C P
Path Cost 0 140 118 75 146 229 239 220 336 317
UCS
140 118 75
239 220
336 317
146 229
299
Explored set: A, Z, T, S, R, L
Fringe A S T Z T L F R C P M
Path Cost 0 140 118 75 146 229 239 220 336 317 299
UCS
140 118 75
239 220
336 317
146 229
299
Explored set: A, Z, T, S, R, L
Fringe A S T Z T L F R C P M
Path Cost 0 140 118 75 146 229 239 220 336 317 299
When does this end?
UCS
140 118 75
239 220
336 317
146 229
299
Explored set: A, Z, T, S, R, L
Fringe A S T Z T L F R C P M
Path Cost 0 140 118 75 146 229 239 220 336 317 299
When does this end? – when the goal state is removed from the queue
UCS
140 118 75
239 220
336 317
146 229
299
Explored set: A, Z, T, S, R, L
Fringe A S T Z T L F R C P M
Path Cost 0 140 118 75 146 229 239 220 336 317 299
When does this end? – when the goal state is removed from the queue
– NOT when the goal state is expanded
UCS
UCS Properties
Is UCS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of UCS? – how many states are expanded before finding a solution?
– b: branching factor – C*: cost of optimal solution – e: min one-step cost – complexity =
What is the space complexity of BFS? – how much memory is required?
– complexity = Is BFS optimal? – is it guaranteed to find the best solution (shortest path)?
Strategy: expand cheapest node first:
Fringe is a priority queue (priority: cumulative cost)
UCS vs BFS
S
a
b
d p
a
c
e
p
h
f
r
q
q c G
a
q e
p
h
f
r
q
q c G
a
3 9 1
16 4 11
5
7 13
8
10 11
17 11
0
6 Cost contours
UCS vs BFS
S
a
b
d p
a
c
e
p
h
f
r
q
q c G
a
q e
p
h
f
r
q
q c G
a
Search
Tiers
Strategy:expandashallowestnodefirst
Implementa6on:FringeisaFIFOqueue
UCS vs BFS
Remember: UCS explores increasing cost contours
The good: UCS is complete and optimal!
The bad: Explores options in every “direction” No information about goal location
We’ll fix that soon!
Start Goal
…
c3 c2
c1
Depth First Search (DFS)
DFS
fringe Fringe A
DFS
fringe
Fringe A B C
DFS
fringe
Fringe A B C F G
DFS
Fringe A B C F G H I
DFS
Fringe A B C F G H I
Which state gets removed next from the fringe?
DFS
Fringe A B C F G H I
Which state gets removed next from the fringe? What kind of a queue is this?
DFS
Fringe A B C F G H I
Which state gets removed next from the fringe? What kind of a queue is this? LIFO Queue!
(last in first out)
Deep/Shallow Water --- DFS, BFS, or UCS? (part 1)
Deep/Shallow Water --- DFS, BFS, or UCS? (part 2)
Deep/Shallow Water --- DFS, BFS, or UCS? (part 3)
DFS Properties: Graph search version
Is DFS complete? – only if you track the explored set in memory What is the time complexity of DFS (graph version)? – how many states are expanded before finding a solution?
– complexity = number of states in the graph What is the space complexity of DFS (graph version)? – how much memory is required?
– complexity = number of states in the graph Is DFS optimal? – is it guaranteed to find the best solution (shortest path)?
This is the “graph search” version of the algorithm
DFS Properties: Graph search version
Is DFS complete? – only if you track the explored set in memory What is the time complexity of DFS (graph version)? – how many states are expanded before finding a solution?
– complexity = number of states in the graph What is the space complexity of DFS (graph version)? – how much memory is required?
– complexity = number of states in the graph Is DFS optimal? – is it guaranteed to find the best solution (shortest path)?
This is the “graph search” version of the algorithm
So why would we ever use this algorithm?
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
This is why we might want to use DFS
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity =
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete?
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete? NO!
DFS: Tree search version
Suppose you don't track the explored set. – why wouldn't you want to do that?
This is the “tree search” version of the algorithm
What is the space complexity of DFS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete? NO!
What do we do???
IDS: Iterative deepening search
What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage
IDS: Iterative deepening search
What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage What is depth limited search? – any guesses?
IDS: Iterative deepening search
What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage What is depth limited search? – do DFS up to a certain pre-specified depth
IDS: Iterative deepening search
§ Idea:getDFS’sspaceadvantagewithBFS’s6me/shallow-solu6onadvantages§ RunaDFSwithdepthlimit1.Ifnosolu6on…
§ RunaDFSwithdepthlimit2.Ifnosolu6on…
§ RunaDFSwithdepthlimit3.…..
§ Isn’tthatwastefullyredundant?§ Generallymostworkhappensinthelowestlevelsearched,sonotsobad!
IDS
IDS
What is the space complexity of IDS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete?
IDS
What is the space complexity of IDS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete? YES!!! Is it optimal?
IDS
What is the space complexity of IDS (tree version)? – how much memory is required?
– b: branching factor – m: maximum depth of any node – complexity =
What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution?
– complexity = Is it complete? YES!!! Is it optimal? YES!!!
TheOneQueue
§ Allthesesearchalgorithmsarethesameexceptforfringestrategies§ Conceptually,allfringesarepriorityqueues(i.e.collec6onsofnodeswithaTachedpriori6es)
§ Prac6cally,forDFSandBFS,youcanavoidthelog(n)overheadfromanactualpriorityqueue,byusingstacksandqueues
§ Canevencodeoneimplementa6onthattakesavariablequeuingobject
SearchandModels
§ Searchoperatesovermodelsoftheworld§ Theagentdoesn’tactuallytryalltheplansoutintherealworld!
§ Planningisall“insimula6on”
§ Yoursearchisonlyasgoodasyourmodels…
SearchGoneWrong?